185 research outputs found
A functional quantum programming language
We introduce the language QML, a functional language for quantum computations
on finite types. Its design is guided by its categorical semantics: QML
programs are interpreted by morphisms in the category FQC of finite quantum
computations, which provides a constructive semantics of irreversible quantum
computations realisable as quantum gates. QML integrates reversible and
irreversible quantum computations in one language, using first order strict
linear logic to make weakenings explicit. Strict programs are free from
decoherence and hence preserve superpositions and entanglement - which is
essential for quantum parallelism.Comment: 15 pages. Final version, to appear in Logic in Computer Science 200
Pure Nash Equilibria: Hard and Easy Games
We investigate complexity issues related to pure Nash equilibria of strategic
games. We show that, even in very restrictive settings, determining whether a
game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has
a strong Nash equilibrium is SigmaP2-complete. We then study practically
relevant restrictions that lower the complexity. In particular, we are
interested in quantitative and qualitative restrictions of the way each players
payoff depends on moves of other players. We say that a game has small
neighborhood if the utility function for each player depends only on (the
actions of) a logarithmically small number of other players. The dependency
structure of a game G can be expressed by a graph DG(G) or by a hypergraph
H(G). By relating Nash equilibrium problems to constraint satisfaction problems
(CSPs), we show that if G has small neighborhood and if H(G) has bounded
hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and
Pareto equilibria is feasible in polynomial time. If the game is graphical,
then these problems are LOGCFL-complete and thus in the class NC2 of highly
parallelizable problems
Diagrammatic Semantics for Digital Circuits
We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about parametrised circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications
The Minimum Oracle Circuit Size Problem
We consider variants of the minimum circuit size problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MSCP[superscript QBF] is known to be complete for PSPACE under ZPP reductions. We show that it is not complete under logspace reductions, and indeed it is not even hard for TC[superscript 0] under uniform AC[superscript 0] reductions. We obtain a variety of consequences that follow if oracle versions of MCSP are hard for various complexity classes under different types of reductions. We also prove analogous results for the problem of determining the resource-bounded Kolmogorov complexity of strings, for certain types of Kolmogorov complexity measures.National Science Foundation (U.S.) (grants CCF-1064785, CCF-1423544, and CCF-1555409)Natural Sciences and Engineering Research Council of Canada (Discovery Grant
From Ultrafilters on Words to the Expressive Power of a Fragment of Logic
International audienceWe give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving profinite equations characterizing B ∩ Reg via ultrafilter equations satisfied by B. This suffices to establish the decidability of the membership problem for B ∩ Reg.Nous donnons une méthode pour spécifier des équations en ultrafiltres et identifier leurs projections sur l'ensemble des mots profinis. Soit B l'ensemble des langages décrits par des énoncés du premier ordre utilisant des prédicats unaires pour chaque lettre, des prédicats numériques uniformes unaires arbitraires et un prédicat pour la longueur d'un mot. Nous illustrons notre méthode en donnant des équations profinies caractérisant B ∩ Reg à partir d'équations en ultrafiltres satisfaites par B. Cela suffit à établir la décidabilité de l'appartenance à B ∩ Reg
Feasible set functions have small circuits
The Cobham Recursive Set Functions (CRSF) provide an analogue of polynomial time computation which applies to arbitrary sets. We give three new equivalent characterizations of CRSF. The first is algebraic, using subset-bounded recursion and a form of Mostowski collapse. The second is our main result: the CRSF functions are shown to be precisely the functions computed by a class of uniform, infinitary, Boolean circuits. The third is in terms of a simple extension of the rudimentary functions by transitive closure and subset-bounded recursion
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